CHAPTER 7: ESTIMATIONREVIEW

Estimation

Estimation is a type of inference where we estimate the of a population parameter (6.4 page 292)

There are two ways to estimate1. Use a point estimate 2. Use a confidence interval

Point Estimate

A point estimate of a population parameter is a single number estimate of the parameter. is the point estimate for μ is the point estimate for is the point estimate for is the point estimate for

Confidence Intervals

A confidence interval is an interval estimate of a population parameter and is used to indicate the reliability of an estimate. General Format of a confidence interval:

point estimate – < parameter < point estimate +

the maximal margin of error (the value of E depends on the situation)

the confidence level is the reliability of an estimate Critical Values are based on the confidence level

and are either or depending on the situation

Confidence Interval for μ when σ is Known

RequirementsLet x be a random variable appropriate to your application. Obtain

a simple random sample (of size n) of x values from which you compute the sample mean . The value of σ is already known (perhaps from a previous study).If you can assume that x has a normal distribution, then any sample size n will work. If you cannot assume this, then use a sample size of n ≥ 30.

Confidence Interval for μ when σ is known:

Where sample mean of a simple random sample

Confidence level

Critical Value for confidence level c

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How To Construct a Confidence Interval for μ when σ is Known1. Check Requirements

Simple random sample? Assumption of normality? Sample size? Sample mean? Population standard deviation σ?

2. Compute 3. Construct the interval using

Confidence Interval for μ when σ is Unknown

RequirementsLet x be a random variable appropriate to your application. Obtain a simple random sample (of size n) of x values from which you compute the sample mean and the sample standard deviation s.

If you can assume that x has a normal distribution or is mound-shaped, then any sample size n will work.

If you cannot assume this, then use a sample size of n ≥ 30. Confidence Interval for μ when σ is unknown

where

= sample mean of a simple random sample

= confidence level (0 < c < 1)

= critical value

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d.f. = n – 1

How To Construct a Confidence Interval1. Check Requirements

Simple random sample? Assumption of normality? Sample size? Sample mean? Sample standard deviation s?

2. Compute E3. Construct the interval using

How to Find a Confidence Interval for a Proportion

Requirements Consider a binomial experiment with trials, where

represents the population probability of success on a single trial and represents the population probability of failure. Let be a random variable that represents the number of successes out of the binomial trials.

The point estimates for and are and The number of trials n should be sufficiently large so

that both and Confidence Interval for :

confidence level critical value based on

How To Construct a Confidence Intervalfor a Proportion

1. Check Requirements Binomial experiment? Compute ? ?

2. Compute 3. Construct the interval using

Confidence Intervals for the difference between two population parameters

There are several types of confidence intervals for the difference between two population parameters Confidence Intervals for 1 – 2 (1 and 2

known) Confidence Intervals for 1 – 2 (1 and 2

Are Unknown) Confidence Intervals for 1 – 2 (1 = 2) Confidence Intervals for p1 – p2

Suppose we construct a c% confidence interval for (or ), then 3 cases arise:1. The confidence interval contains only negative values

We conclude that (or ) We state: “we are c% confident that (or )”

2. The confidence interval contains only positive values We conclude that (or ) We state: “we are c% confident that (or )”

3. The confidence interval contains both positive and negative values

In this case we cannot conclude if or (or or) is larger. A shorter confidence interval might put us back to case 1 or 2

If we reduce the confidence level, then the confidence interval will be shorter.

If we increase the sample size, then the confidence interval will be shorter.

How to Interpret Confidence Intervals for Differences

RequirementsLet and be the population standard deviations of population 1 and 2. Obtain two independent random samples from population 1 and 2, where

and are sample means from population 1 and 2

and are sample sizes from population 1 and 2

If you can assume that both population distributions are normal, any sample size will work.If you cannot assume this, then the sample sizes must be greater than or equal to 30.

Confidence Intervals for 1 – 2

(1 and 2 known)

Confidence Interval for

Where

confidence level critical value for the confidence level

Confidence Intervals for 1 – 2

(1 and 2 known)

How to construct the confidence interval for 1 – 2 (1 and 2 known)

1. Check Requirements Are and known? Are the samples independent? Assumption of normality?

2. Compute E: 3. Construct the Interval:

RequirementsObtain two independent random samples from population 1 and 2, where

and are sample means from population 1 and 2 and are sample standard deviations from population 1 and 2

and are sample sizes from population 1 and 2

If you can assume that both population distributions are normal, any sample size will work.If you cannot assume this, then the sample sizes must be greater than or equal to 30.

Confidence Intervals for 1 – 2

(1 and 2 Are Unknown)

Confidence Interval for

Where

confidence level critical value for the confidence level d.f. = smaller of and

Confidence Intervals for 1 – 2

(1 and 2 Are Unknown)

How to construct the confidence interval for 1 – 2 (1 and 2 unknown)

1. Check Requirements Are and known? Are the samples independent? Assumption of normality?

2. Compute E: Note: d.f. = smaller of and

3. Construct the Interval:

RequirementsConsider two independent binomial experiments

Estimating the Difference ofProportions p1 – p2

Binomial Experiment 1 Number of trials number of successes out of trials population probability of success

Binomial Experiment 2 Number of trials number of successes out of trials population probability of success

The number of trials should be sufficiently large so that the following are true:

Confidence Interval for

Where confidence level critical value

Estimating the Difference ofProportions p1 – p2

How to construct the confidence interval for p1 – p2

1. Check Requirements Two independent binomial experiments

2. Compute E: 3. Construct the interval:

How to Find the Sample Size n for Estimating μ when is σ known

Requirements The distribution is approximately normal

Formula for Sample Size

If is not a whole number, increase to the next higher whole number.

Note: is the minimal sample size for a specified confidence level and maximal error of estimate.

If n ≥ 30, then it is usually safe to use instead of σ

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Sample Size for Estimating

Two ways to find the sample size1. if you have a preliminary estimate for 2. if you do not have a preliminary estimate for Specified maximal error of estimate critical value from the normal distribution for the desired confidence level If is not a whole number, increase to the next whole number

If necessary, increase to ensure 5 and

Recall that is the minimal sample size for a specified confidence level and maximal error of estimate (it is OK to use a larger )

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